Quasi-steady state of hub vortex under multi-polar strain caused by three satellite vortices

The goal of our study is to investigate the stability and the non-linear dynamics of the hub vortex, which is a prominent feature of wakes shed by rotating bodies such as windmills. The hub vortex is accompanied by helical vortices, and both interact with each other. While some studies have been devoted to the dynamics and stability of helical vortices, the investigation into the former is comparatively limited. Consequently, as an initial approach, we conduct both theoretical and numerical analyses to achieve a two-dimensional quasi-steady state of the hub vortex, considering its interaction with a multi-polar strain field caused by three satellite vortices. In the theoretical analysis, we perform an asymptotic expansion of vorticity and stream function up to the second order in ε, which represents the strength of the straining field induced by the satellite vortices. This allows us to obtain the basic flow of the strained hub vortex. This solution is then carefully compared to the quasi-steady states obtained using direct numerical simulations. Comparisons reveal a clear similarity between the basic flow obtained through theoretical and numerical approaches. Nevertheless, before proceeding with the subsequent linear stability analysis, a more thorough examination is required.